The column-sufficiency and row-sufficiency of the linear transformation on Hilbert spaces

Given a real Hilbert space H with a Jordan product and being the Lorentz cone, , and let T : H → H be a bounded linear transformation, the corresponding linear complementarity problem is denoted by LCP( T , Ω, q ). In this paper, we introduce the concepts of the column-sufficiency and row-sufficienc...

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Bibliographic Details
Published inJournal of global optimization Vol. 49; no. 1; pp. 109 - 123
Main Authors Miao, Xin-He, Huang, Zheng-Hai
Format Journal Article
LanguageEnglish
Published Boston Springer US 2011
Springer Nature B.V
Subjects
Computer Science
Hilbert space
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Real Functions
Lorentz cone
Linear complementarity problem
Row-sufficiency
Jordan product
Column-sufficiency
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Summary:Given a real Hilbert space H with a Jordan product and being the Lorentz cone, , and let T : H → H be a bounded linear transformation, the corresponding linear complementarity problem is denoted by LCP( T , Ω, q ). In this paper, we introduce the concepts of the column-sufficiency and row-sufficiency of T . In particular, we show that the row-sufficiency of T is equivalent to the existence of the solution of LCP( T , Ω, q ) under an operator commutative condition; and that the column-sufficiency along with cross commutative property is equivalent to the convexity of the solution set of LCP( T , Ω, q ). In our analysis, the properties of the Jordan product and the Lorentz cone in H are interconnected.
ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-010-9537-5